arXiv subjects

Jérôme Carrand

Publications and source records attributed to Jérôme Carrand.

6 recordsLinked to original sources

arXiv2026

Entropy structures with continuous partitions of unity

Using only continuous partitions of unity, we provide equivalent definitions for the metric, topological and topological tail entropies and pressures of a continuous self-map of a compact set, as well as their conditional versions. A tail variational principle for these new definitions is proved. We extend Downarowicz's notions of candidates and entropy structures to account for almost-increasing sequences of functions arising from the new definitions. Finally, we deduce a partial answer to a question raised by Newhouse.

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arXiv2024

A direct proof of the existence of MME for finite horizon Sinai billiards

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gaz, is a discontinuous map. Assuming finite horizon and another condition we introduce -- namely \emph{negligible singularities} -- we prove that the metric pressure map associated with the billiard map $T$ is upper semi-continuous, as well as the compactness of the set of $T$-invariant measures. In particular, for the potentials $g \equiv 0$ and $g = -h_{\rm top}(Φ^1) τ$, we recover the recent results of the existence of measures of maximal entropy (MME) for both the billiard map and flow $Φ^t$, due to Baladi and Demers for $T$, jointly with Carrand for $Φ^t$. We prove that the negligible singularities condition is generic among the billiard table with $C^{3+α}$ boundary, with respect to the $C^{3+α}$ topology. For finite horizon Sinai billiards, we provide bounds on the defect of upper semi-continuity of the metric pressure map and on the topological tail entropy. Assuming that singularities are negligible and sparse recurrence hold, we deduce the equidistribution of the periodic orbits with respect to the unique MME. We provide examples of billiard tables satisfying both conditions.

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arXiv2024

A family of natural equilibrium measures for Sinai billiard flows

The Sinai billiard flow on the two-torus, i.e., the periodic Lorentz gas, is a continuous flow, but it is not everywhere differentiable. Assuming finite horizon, we relate the equilibrium states of the flow with those of the Sinai billiard map $T$ -- which is a discontinuous map. We propose a definition for the topological pressure $P_*(T,g)$ associated to a potential $g$. We prove that for any piecewise Hölder potential $g$ satisfying a mild assumption, $P_*(T,g)$ is equal to the definitions of Bowen using spanning or separating sets. We give sufficient conditions under which a potential gives rise to equilibrium states for the Sinai billiard map. We prove that in this case the equilibrium state $μ_g$ is unique, Bernoulli, adapted and gives positive measure to all nonempty open sets. For this, we make use of a well chosen transfer operator acting on anisotropic Banach spaces, and construct the measure by pairing its maximal eigenvectors. Last, we prove that the flow invariant probability measure $\bar μ_g$, obtained by taking the product of $μ_g$ with the Lebesgue measure along orbits, is Bernoulli and flow adapted. We give examples of billiard tables for which there exists an open set of potentials satisfying those sufficient conditions.

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arXiv2024

Measure of maximal entropy for finite horizon Sinai billiard flows

Using recent work of Carrand on equilibrium states for the billiard map, and bootstrapping via a "leapfrogging" method from a previous article of Baladi and Demers, we construct the unique measure of maximal entropy for two-dimensional finite horizon Sinai (dispersive) billiard flows (and show it is Bernoulli), assuming that the topological entropy of the flow is strictly larger than s_0 log 2 where 0<s_0<1 quantifies the recurrence to singularities. This bound holds in many examples (it is expected to hold generically).

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arXiv2022

Smooth Generalized Interval Exchange Transformations with Wandering Intervals, from explicit Derived from pseudo-Anosov maps

Starting from any pseudo-Anosov map $φ$ on a surface of genus $g \geqslant 2$, we construct explicitly a family of Derived from pseudo-Anosov maps $f$ by adapting the construction of Smale's Derived from Anosov maps on the two-torus. This is done by perturbing $φ$ at some fixed points. We first consider perturbations at every conical fixed point and then at regular fixed points. We establish the existence of a measure $μ$, supported by the non-trivial unique minimal component of the stable foliation of $f$, with respect to which $f$ is mixing. In the process, we construct a uniquely ergodic Generalized Interval Exchange Transformation with a wandering interval that is semi-conjugated to a self-similar Interval Exchange Transformation. This Generalized Interval Exchange Transformation is obtained as the Poincaré map of a flow renormalized by $f$ which parametrizes stable foliation. When $f$ is $\mathcal{C}^2$, the flow and the Generalized Interval Exchange Transformation are~$\mathcal{C}^1$.

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arXiv2022

Logarithmic bounds for ergodic sums of certain flows on the torus: a short proof

We give a short proof that the ergodic sums of $\mathcal{C}^1$ observables for a $\mathcal{C}^1$ flow on $\mathbb{T}^2$ admitting a closed transversal curve whose Poincaré map has constant type rotation number have growth deviating at most logarithmically from a linear one. For this, we relate the latter integral to the Birkhoff sum of a well-chosen observable on the circle and use the Denjoy-Koksma inequality. We also give an example of a nonminimal flow satisfying the above assumptions.

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